91Ó°ÊÓ

Understanding variable separation is crucial when solving differential equations. It’s a technique used to separate the variables in an equation so that each side of the equation contains only one variable. In our given problem, we start with: \( \frac{d y}{d t} = -\frac{1}{t^{2} y^{2}} \). To separate the variables, we need to place all terms involving 'y' on one side and all terms involving 't' on the other side. Multiplying both sides by \( y^2 \, dt \), we get: \[ y^2 \, dy = -\frac{1}{t^2} \, dt \]. Now, we have successfully separated the variables, preparing us for the next step: integration.

Integration is the next crucial step after separating the variables. It involves finding the antiderivative of both sides of the equation. For our problem, we have: \[ \begin{aligned} \int y^2 \, dy &= \int -\frac{1}{t^2} \, dt \end{aligned} \].
Evaluate the integral on the left side first: \[ \int y^2 \, dy = \frac{y^3}{3} \].
Now, evaluate the integral on the right side: \[ \int -\frac{1}{t^2} \, dt = \- \frac{1}{t} \]. Integration introduces a constant of integration, 'C', so we have: \[ \frac{y^3}{3} = -\frac{1}{t} + C \]. These evaluated integrals allow us to solve for 'y'.

The final step in solving our differential equation is to isolate 'y' and express it in terms of 't'. From the integrated equation, \[ \frac{y^3}{3} = -\frac{1}{t} + C \], multiply both sides by 3 to simplify: \[ y^3 = -\frac{3}{t} + 3C \].
Next, take the cube root of both sides to solve for 'y': \[ y = \sqrt[3]{-\frac{3}{t} + 3C} \].
This result is the general solution to the original differential equation. Remember, solving differential equations allows us to understand how variables change in relation to each other. Breaking the problem into clear, manageable steps—like separation of variables, integration, and finally solving the equation—makes the process more approachable and less daunting.